HAL Id: insu-01100866
https://hal-insu.archives-ouvertes.fr/insu-01100866
Submitted on 7 Jan 2015HAL is a multi-disciplinary open access archive for the deposit and dissemination of sci-entific research documents, whether they are pub-lished or not. The documents may come from teaching and research institutions in France or abroad, or from public or private research centers.
L’archive ouverte pluridisciplinaire HAL, est destinée au dépôt et à la diffusion de documents scientifiques de niveau recherche, publiés ou non, émanant des établissements d’enseignement et de recherche français ou étrangers, des laboratoires publics ou privés.
Fictitious domain method for acoustic waves through a
granular suspension of movable rigid spheres
David Imbert, Sean Mcnamara, Yves Le Gonidec
To cite this version:
David Imbert, Sean Mcnamara, Yves Le Gonidec. Fictitious domain method for acoustic waves through a granular suspension of movable rigid spheres. Journal of Computational Physics, Elsevier, 2015, 280, pp.676-691. �10.1016/j.jcp.2014.10.006�. �insu-01100866�
Fictitious domain method for acoustic waves through a granular
suspension of movable rigid spheres
D. Imberta,c,1, S. McNamaraa,c,∗, Y. Le Gonidecb,c
aInstitut de Physique de Rennes (UMR Universit´e de Rennes 1/ CNRS 6251) bG´eosciences Rennes (UMR Universit´e de Rennes 1/ CNRS 6118)
cCampus de Beaulieu, 263 av. du G´en´eral Leclerc, CS 74205, 35042 RENNES CEDEX, FRANCE
Abstract
We develop a model to couple acoustic waves and the motion of rigid movable grains in a sub-merged suspension. To do so, we use the fictitious domain method based on distributed Lagrange multipliers to enforce the natural jump condition of the wave equation and a rigidity constraint. One can then model the granular medium with ”Molecular Dynamics” or related methods. Both dynamic and acoustic numerical results are compared with analytic solutions of acoustics and an estimation of the error is given. We show energy transfers between two submerged grains sepa-rated by water and linked to their own spring while oscillating. The model is applied at large scale with a suspension of many grains to foresee its future possibilities.
Keywords: granular medium, suspension, fictitious domain method, distributed Lagrange
multipliers, movable rigid grains, acoustic wave equation
1. Introduction
In marine Geosciences, acoustic probing is commonly used to detect subsurface reflectors, characterize their physical properties and study geophysical processes such as slope instabilities. In particular, undersea sediment layers constitute granular materials which pore space is filled with waver, i. e. grains are movable bodies. Such layers are acoustically complex materials where sound waves can be multi-scattered and reflected at fluid-grain interfaces. Experiments with model spherical grains under water have been performed to better understand the acoustic behavior of granular media in equilibrium [1] or tilted under gravity [2, 3]. These experiments clearly highlight the contribution of the fluid-grain interaction on the acoustic measurements, which can also be affected by inter-particle contacts [4].
In the case of pores filled with air, dry granular materials are commonly modeled using discrete element methods such as ”Molecular Dynamics” [5, 6]. Each grain is represented using a rigid and
✩This work was supported by the ANR project STABINGRAM No. 2010-BLAN-0927-01.
∗Corresponding author.
Email addresses: david.imbert.1@free.fr (D. Imbert), sean.mcnamara@univ-rennes1.fr
(S. McNamara), yves.legonidec@univ-rennes1.fr (Y. Le Gonidec)
1D. Imbert’s work was supported by the grant No. 211-14-231/ARED from the R´egion Bretagne.
incompressible sphere completely characterized by the position of its center and its radius. The air filling the pores can be neglected because its density is very low, so that the momentum it transmits is negligible. In the case of pores filled with water, the momentum transmitted by the fluid can no longer be neglected. Thus, discrete element methods can not be used to model submerged granular media. Well known numerical models exist to model sound waves in a liquid: for example finite differences [7, 8] or finite elements [9, 8]. But to our knowledge, none of them usually consider neither the coupling between the wave and the motion of grains, nor the inter-particles contacts.
The fictitious domain method based on Lagrange multipliers was introduced in [10] and started to be popular two decades ago when applied to Navier-Stokes equations [11] and then for scatter-ing problems in acoustics [12]. The particularity of this method is to work with two independent finite element meshes: the first one regular and time-independant and the second one unstruc-tured and possibly time-dependant. This last attribute was first applied for forced displacement of particles in [13] and to model vibrations of the membrane of a kettledrum [14]. Introduction of distributed Lagrange multipliers [15] allowed simulation of freely movable particles through a rigid body motion constraint applied inside particles[16]. The fictitious domain method based on distributed Lagrange multipliers was also developed in [17] for the mixed formulation of the acoustic wave equation with Dirichlet conditions.
In this work, we unite a finite element method to a discrete element model through the fictitious domain method, yielding a single method capable of describing acoustic propagation by both the fluid and inter-granular contacts.
To our knowledge, our model introduced in [18] is the first to couple the acoustic wave equa-tion governing the fluid to the dynamic equaequa-tions governing the moequa-tion of rigid movable grain. However, in that work, we simulated a single rigid sphere in a sound wave, and we compared only its velocity to the analytic solution. In addition to results on the dynamic of the grain, our model allows each grain to generate its own acoustic field when it is under a force due to another incident wave but also due to any other forces acting on it such as a contact force or the gravity. These resulting acoustic fields have never been validated and confronted to analytic solutions.
In the present paper, we present a major development of the model introduced in [18]. First of all, we take on a new mixed finite elements method based on standard conforming finite elements to fix severe convergence issues of the fictitious domain method. We also change the numerical scheme to improve energy conservation. Finally, the model is now able to work with several grains and we illustrate that with new numerical experiments.
The paper is subdivided into 2 main parts. In section 2 and 3, we are describing the model, i. e. governing equations, the fictitious domain method, and the computational scheme. Section 4 deals with three numerical experiments, involving one, two and many movable grains submerged in water, respectively. With the first experiment, both dynamic and acoustic numerical results are compared with analytic solutions of acoustics and an estimation of the error is given. With the second experiment, we analyse the energy transfers between two oscillating grains separated by water and linked with their own springs. In the last experiment, we highlight the potential of the model to sound a suspension of grains with acoustic pulses.
Ω Γ1 Γ2 Γ3 Γ4 G1 ∂G1 G2 ∂G2 G3 ∂G3 n1 t1 n2 t2 n3 t3 ex ey nΓ
Figure 1: System with the different domains: domains Gk are the domain of each grain k ; and the hatched rectangle is the whole domainΩ and Γ its boundary.
2. Governing equations
A schematic representation of rigid bodies in a fluid matrix is considered on Fig. 1. LetΩ be the domain which includes a set of grains G and a fluid matrixΩ \ G. The domain G is subdivided into NGcircular subdomains Gk, one for each grain, such that G= ∪ Gk. LetΓ be the boundary of the whole domainΩ, ∂Gk the boundary of each domain Gk, and therefore let∂G be the boundary of the domain G. We also define nk the outgoing normal unit vector of the boundary∂Gk, tk the tangential unit vector, and Xkthe position of the center of the subdomain Gk.
We consider that the fluid is perfect, i.e. non viscous, non heat conducting and isentropic. The perfect fluid approximation is commonly used in acoustics because the scatteering and transport of energy are much more important than dissipation due to viscosity. Moreover, the variations of the pressure fields are small compared to their mean value and the density of the fluid ρ0 and
the velocity of sound in the fluid c0 are constant. These assumptions allow use of the mixed
formulation of the acoustic wave equation governing the fluid.
2.1. Strong forms
We can write the acoustic wave equation as a system of two equations in terms of pressure field p and velocity field u:
⎧⎪⎪ ⎪⎪⎪⎪⎪ ⎨ ⎪⎪⎪⎪⎪ ⎪⎪⎩ ρ0∂u ∂t + ∇ p = 0 in Ω \ G 1 ρ0c20 ∂p ∂t + ∇ · u = 0 in Ω \ G (1a) (1b)
The second system of equations governs the motion of grains: ρ Sk dUk dt = − ∂Gk p nkd + Fk ∀ k = 1, 2, ..., NG (2) 3
Here Skand Ukare the surface (in a 2D domain) and the velocity of the grain k, respectively. All grains have the same density ρk = ρ ∀ k = 1, 2, ..., NG. Fk can handle any other extra force, for example if the grain is subjected to the gravity, a spring force, or some interaction forces. Contrary to system (1), Eq. (2) is discrete and therefore allows use of discrete elements methods such that the Molecular Dynamics algorithm [5].
Then, to link systems (1) and (2), we add the natural jump condition of the acoustic wave equation for movable rigid bodies:
u· nk = Uk· nk on∂Gk∀ k = 1, 2, ..., NG (3)
Eq. (3) is essential to couple the grains velocity Uk and the acoustic velocity field u.
Eq. (3) is just the ”slip” boundary condition appropriate for the perfect fluid we are consider-ing, applied to the surface of the grains. The particle rotations do not appear because they couple to the fluid only through tangential viscous stresses exerted by the viscosity of the particle surface. Our fluid is inviscid and therefore this coupling is absent.
Note that the jump condition (3) does not prevent particle rotation. In the special case where Fk includes friction forces between grains, then an additional term including angular velocity should be added to the first member of Eq. (2). But even in this case, rotations of a sphere in a perfect fluid can not generate any sound.
Another natural consequence of Eq. (3) is that only the normal component of the acoustic velocity field is constrained at the interface. Because of the perfect fluid, we have no information on the tangential component of the velocity field. This is consistent with the governing equation Eq. (1) which allows more freedom for the velocity and pressure fields than the Navier-Stokes equation would do.
Two kinds of boundary conditions are set on the outer frontierΓ = ∪ Γj ofΩ. On one hand, Perfectly Matched Layers (PML) [19] are set on the bottom side Γ1 and the top side Γ3 of the
domainΩ but we will not focus on that point to keep equations simple. See [20] for details on the construction of the PML based on a coordinate stretching approach for Maxwell’s equations and [17] for the acoustic wave equation. These layers end with a Dirichlet condition applied on the pressure field:
p = 0 on Γ1∪ Γ3 (4)
On the other hand, we set periodic boundary conditions on the right sideΓ2and the left sideΓ4of
the domainΩ
p|Γ2 = p|Γ4 onΓ2∪ Γ4 (5)
u|Γ2 = u|Γ4 onΓ2∪ Γ4 (6)
In that way, we can easily generate plane waves which are useful when comparing with analytic solutions of acoustics.
2.2. Weak forms
We write now the weak form of Eqs. (1a) and (2) into a single equation with the unknowns
u∈ H(div, Ω \ G), Uk∈ R2∀ k = 1, ..., NG and p∈ L2(Ω \ G): Ω\Gρ0 ∂u ∂t · v dx − Ω\G p∇ · v dx + Γ p nΓ· v d − ∂G p n· v d + k ρ Sk dUk dt · Vk− Fk· Vk+ ∂Gk p nk· Vkd = 0 (7) Here v∈ H(div; Ω \ G)2, V
k∈ R2, and q∈ L2(Ω \ G) are the respective test functions of u, Uk, and
p. Eq. (7) was called combined equation of motion in [21] and it expresses the conservation of the
momentum in the whole domainΩ. We note that the integral on ∂G (last term of the 1st line of (7)) and the sum of hydrodynamic forces (last term of the 2nd line of (7)) cancel each other out if and only if we extend Eq. (3) to the test functions and require v· nk = Vk· nk∀ k = 1, 2, ..., NG. To take this condition into account, we define a new functional space ˜WΩ\G that we also called combined
variation space in reference to the space introduced in [16] for the Navier-Stokes equation:
˜ WΩ\G = (v, Vk) v∈ H(div; Ω \ G), Vk ∈ R2; v· nk = Vk· nk on∂Gk ∀k = 1, 2, ..., NG (8) Moreover, boundary conditions (4) and (5) make the integral on Γ vanish, and therefore lead to find (u, Uk)∈ ˜WΩ\Gand p∈ L2(Ω \ G) such as:
⎧⎪⎪ ⎪⎪⎪⎪⎪ ⎪⎨ ⎪⎪⎪⎪⎪ ⎪⎪⎪⎩ Ω\Gρ0 ∂u ∂t · v dx − Ω\G p∇ · v dx + k ρ Sk dUk dt · Vk− Fk· Vk = 0 Ω\G 1 ρ0c20 ∂p ∂t q dx+ Ω\G∇ · u q dx = 0 (9a) (9b) ∀ (v, Vk)∈ ˜WΩ\Gand q∈ L2(Ω \ G).
2.3. Fictitious domain method formulation
The fictitious domain method consists in extending Eq. (9) defined in the fluid domainΩ \ G to the domainΩ such that the acoustic fields p and u are also defined in the whole regular domain Ω. To do so, we proceed in three steps :
1. first, we extend the combined equation of motion (9a) to the whole domainΩ;
2. then, constraints acting on the grain are enforced in a weak sense using Lagrange multipliers; 3. finally, we extend the linearized continuity equation (9b) to the whole domain using the
same constraints
2.3.1. Combined equation of motion for the extended domain
First, we follow the method described in [16] to develop an equation analogous to Eq. (9a) valid in the domain Gk. The extension from one grain Gk to the whole domain G = Gk is straightforward. However, we start with a different hypothesis from [16] because we use a different boundary condition on the grain, namely Eq. (3).
Let (Xk, er, eθ) be an orthonormal basis with the center of the grain Xk as its origin and er =
nk and eθ = tk orthogonal unit vectors. The following development to find an equation for the extended domain Gkis split into two steps: one to find an expression involving the projection of u on erand another one for the projection of u on eθ.
Firstly Eq. (3) is extended from∂Gkto Gk:
u· er = Uk· er in Gk (10) Let’s call ur the projection of u on er:
ur = Uk· er (11)
We can derive Eq. (11) with respect to the time and multiply each member byρ0:
ρ0∂u
r ∂t = ρ0,
d
dt(Uk· er) in Gk (12)
We introduce the combined variation space ˜WGk, counterpart of the space ˜WΩ\Gin Gk: ˜ WGk = (v, Vk) v∈ H(div; Gk), V ∈ R2; v· nk= Vk· nkon∂Gk (13) The weak form of Eq. (12) can be written:
Gk ρ0∂u r ∂t vrdx= Gk ρ0 d dt(Uk· er) Vkrdx (14)
Then, knowing that er = cos(θ) ex+ sin(θ) ey: Gk ρ0∂u r ∂t vrdx= ρ0 Rk 0 dr 2π 0 d dt Ukx cos(θ) + Uky sin(θ) Vkx cos(θ) + Vky sin(θ) dθ (15) we compute integrals in the right hand side of (15):
Gk ρ0∂u r ∂t vrdx= ρ0Sk dUk dt · Vkdx (16)
Now, we do the same for the tangential component of the velocity field uθ= u · tk(see Fig. 1 for the direction of tk). We begin with another natural jump condition of the wave equation derived from the continuity of the pressure through the interface∂Gk[22]:
ρ0 ∂
∂t(u· tk)= ∇ p · tk on∂Gk (17)
As described above, we extend Eq. 17 to Gk, multiply by the test function vθ and we integrate using the gradient theorem on a closed surface so that:
Gk
ρ0∂uθ
Injecting Eq. (18) in Eq. (16) leads to: Gk ρ0∂u ∂t · v dx = ρ0Sk dUk dt · Vkdx ∀(v, Vk)∈ ˜WGk (19)
To go further, we enforce a divergence free constraint of the velocity field inside the grain Gk to prevent propagation of waves through the grain:
∇ · u = 0 in Gk (20)
The associated constraint∇ · v = 0 related to Eq. (20) has to be joined to the functional space ˜WGk (13): WGk = (v, Vk) v∈ H(div; Gk), Vk∈ R2; v· nk= Vk· nkon∂Gk; ∇ · v = 0 in Gk (21) In that way, we can add the missing integral to obtain the analog of Eq. (9a) and the problem is now to find (u, Uk)∈ WGk such as:
Gk ρ0∂u ∂t · v dx − Gk p∇ · v dx = ρ0Sk dUk dt · Vkdx ∀(v, Vk)∈ WGk (22) Then, performing the symbolic operation ”(9a) +k (22)” yields to the combined equation of motion for the extended domain with solutions (u, Uk)∈ WΩ:
Ωρ0 ∂u ∂t · v dx − Ω p∇ · v dx + k (ρ − ρ0) Sk dUk dt · Vk− Fk· Vk = 0 ∀(v, Vk)∈ WΩ (23) withWΩ the combined variation space extended toΩ:
WΩ =
(v, Vk) v∈ H(div; Ω), Vk∈ R2; v· nk= Vk· nkon∂Gk; ∇ · v = 0 in Gk∀ k = 1, ..., NG (24)
2.3.2. Enforcement of constraints using Lagrange multipliers
In a second phase, the two constraints v· n = V · n and ∇ · v = 0 have to be extracted from the combined variation space (24) and added explicitly to Eq. (23) with Lagrange multipliers. For each unknown Lagrange multiplier added, an equivalent equation have to be joined to Eq. (23). The jump condition of the wave equation (3) is enforced using boundary Lagrange multipliersλ∂Gon ∂G and distributed Lagrange multipliers [15] λGare used to enforce the divergence free constraint (20) inside G. The problem is to find u∈ H(div, Ω), Uk ∈ R2, λ∂G∈ H1/2(∂G), λG ∈ H1(G) such as: ⎧⎪⎪⎪ ⎪⎪⎪⎪⎪ ⎪⎪⎪⎪⎪ ⎪⎪⎪⎪⎪ ⎪⎪⎪⎪⎪ ⎨ ⎪⎪⎪⎪⎪ ⎪⎪⎪⎪⎪ ⎪⎪⎪⎪⎪ ⎪⎪⎪⎪⎪ ⎪⎪⎪⎩ Ωρ0 ∂u ∂t · v dx − Ω p∇ · v dx + k (ρ − ρ0) Sk dUk dt · Vk− Fk· Vk + k ∂Gk λ∂G(v· nk− Vk· nk) d + G λG∇ · v dx = 0 for k= 1, 2, ..., NG ∂Gk (u· nk− Uk· nk)μ∂Gd = 0 G ∇ · u μGdx= 0 ∀v ∈ H(div, Ω), Vk∈ R2, μ∂G∈ H1/2(∂G), μG ∈ H1(G) (25a) (25b) (25c) 7
withμG andμ∂Gtest functions ofλ∂GetλGrespectively.
2.3.3. Linearized continuity equation for the extended domain
Now, let us include Eq. (9b) into the system (25). We write the weak form of the divergence
free constraint (20):
G
∇ · u q dx = 0 ∀q ∈ L2(G) (26)
Note that Eq. (26) is redundant with Eq. (25c). Consequently, we can directly extend Eq. (9b) toΩ because Eq. (25c) is enforced anyway. The final problem consists in finding u∈ H(div, Ω), Uk ∈ R2, p ∈ L2(Ω), λ
∂G ∈ H1/2(∂G), and λG ∈ H1(G) such as: ⎧⎪⎪ ⎪⎪⎪⎪⎪ ⎪⎪⎪⎪⎪ ⎪⎪⎪⎪⎪ ⎪⎪⎪⎪⎪ ⎪⎪⎪⎪⎪ ⎪⎪⎨ ⎪⎪⎪⎪⎪ ⎪⎪⎪⎪⎪ ⎪⎪⎪⎪⎪ ⎪⎪⎪⎪⎪ ⎪⎪⎪⎪⎪ ⎪⎪⎪⎪⎩ Ωρ0 ∂u ∂t · v dx − Ωp∇ · v dx + k (ρ − ρ0) Sk dUk dt · Vk− Fk· Vk + k ∂Gk λ∂G(v· nk− Vk· nk) d + G λG∇ · v dx = 0 Ω 1 ρ0c20 ∂p ∂t q dx+ Ω∇ · u q dx = 0 for k = 1, 2, ..., NG ∂Gk (u· nk− Uk· nk)μ∂Gd = 0 G ∇ · u μGdx= 0 ∀v ∈ H(div, Ω), Vk ∈ R2, q ∈ L2(Ω), μ∂G ∈ H1/2(∂G), μG∈ H1(G) (27a) (27b) (27c) (27d)
Eq. (27) can be solved only ifρ0 ρ, otherwise the unknown disappears from Eq. (27a). A
similar case occurs in [16] for Navier-Stokes equations but another fictitious domain formulation was found to deal with this specific issue in [23]. We do not treat the case where the density of the fluidρ0is the same as the density of the grain ρ because it is not relevant in the case of our
geophysical applications. Moreover, [23] does not refer to an explicit velocity of the grain Uk which we need for the discrete algorithm we use.
3. Computational scheme 3.1. Finite elements
With the fictitious domain approach, two independent meshes have to be considered: one for the global domainΩ and one for the domain of grains G. A regular grid TΩh (see Fig. 2a) is used for the rectangular domainΩ to allow use of optimization techniques for the computation of the pressure and velocity fields p and u. The second gridTGh (Fig. 2b) refers to the domain of grains
G: this grid may be unstructured but requires specific quadrature formula to perform the numerical
integration on triangles. So we prefer to choose a pseudo-regular grid as in [24] to keep variational spaces simple.
(a)TΩh (b)TGh
Figure 2: The regular grid TΩh (on the left) is the grid where the physical variables u and p are computed and the unstructured gridTGh (on the right) is the domain of the Lagrange multipliers.
Furthermore,TGh is independent from the gridTΩh and can be shifted during the simulation. However, the size of the two meshes cannot be chosen independently; there is a compatibility condition that must be satisfied so that the fictitious domain method converges [10]. Concretely, we choose in our experiments a mesh ratio ofκ ≈ 1.2 which yields results in good agreement with analytic solutions.
We use the free software FreeFEM++ [25] to generate meshes and solve the partial differential equations. In fact, FreeFEM++ works in the same way as Matlab does, i.e. with its own high-level programming scripting language, close to the C language. So it is easy to implement both finite elements and the discrete element method governing the motion of grains. The full access to matrices also allows them to be modified to include constraints of the fictitious domain method.
In [18], we based our finite element approximation on the lowest order Raviart-Thomas ele-ment RT0[26] which is the natural choice to discretize the functional space H(div;Ω). Despite the
validation of the particle velocities in [18], the examination of the acoustic fields pointed out se-rious issues with RT0element. Indeed, convergence issues were also revealed in [27] while using
the fictitious domain method with Neumann boundary condition and the RT0 element. We solve
these problems in the present paper by opting for mixed methods using standard conforming finite elements [28]. Lagrangian finite elements are used to approximate the solutions: the piece-wise constant element P0h for the pressure space L
2(Ω) is defined as: P0h = qh∈ L2(Ωh) ∀K∈ TΩh, qh|K = βK (28) with K a triangular finite element. The continuous piece-wise linear element P1h is used to dis-cretize the velocity space H(div;Ω) in TΩh:
P1h =
vh∈ C0(Ωh)| ∀K ∈ TΩh, vh|K∈ P1
(29) and also the Lagrange multipliers spaces H1(G) inT
Gh and H
1/2(∂G) on T ∂Gh. 9
3.2. Finite-differences time-domain method 3.2.1. Operator splitting scheme
System (27) is too intricate to be solved directly using a standard finite difference scheme. It has to be separated into smaller subproblems using the Marchuk’s first order operator splitting scheme [29]. Consider the problem :
∂ϕ
∂t + A (ϕ) = f and ϕ|t=0= ϕ0 (30)
where A=Nα=1Aα, Aα≥ 0, N ≥ 2 and A is time-independent. The absolutely stable algorithm is, for j= 0, 1, ..., N: ϕj+1 N − ϕj Δt + A1 ϕj+N1= fj+1 1 ... ϕj+1− ϕj+N−1N Δt + AN ϕj+1= fj+1 N (31)
withΔt the time step. To apply this algorithm, the system (27) is subdivided into two steps dis-cussed in detail below:
1. propagate the acoustic wave in the fluid (step 1a); compute position, velocity and forces acting on the grains (step 1b),
2. enforce constraints applied on the grains (step 2).
We compute the acoustic velocity field and grain velocities at the same time because they are linked through the same velocity space, so they both correspond with step 1 of Marchuk’s algo-rithm. However, resolutions of the acoustic wave equation and the equation of motion are totally independent inside step 1.
3.2.2. Step 1a: wave equation with semi-implicit Euler
Firstly, we solve the wave equation system by finding uh∈ P21
h and ph∈ P0h such as: ⎧⎪⎪ ⎪⎪⎪⎪⎪ ⎨ ⎪⎪⎪⎪⎪ ⎪⎪⎩ Ω 1 ρ0c20 ∂ph ∂t qhdx+ Ω∇ · uh qhdx= 0 Ωρ0 ∂uh ∂t · vhdx− Ωph∇ · vhdx= 0 (32a) (32b) ∀ vh∈ P21
hand qh∈ P0h. We can solve successively equations (32a) and (32b) using the following first order semi-implicit Euler scheme :
1. find pn+ 1 2 h ∈ P0hsuch as: Ω 1 ρ0c20 pn+12 h − p n h Δt qhdx+ Ω∇ · u n hqhdx= 0 ∀ qh∈ P0h (33)
2. find un+12 h such as: Ωρ0 un+ 1 2 h − u n h Δt · vhdx− Ω pn+ 1 2 h ∇ · vhdx= 0 ∀ vh∈ P 2 1h (34)
with t = Δtmin. In the semi-implicit Euler scheme, the minimal time step Δtmin is related to the
minimal spatial step hminby:
Δtmin ≤ α
c0
hmin (35)
whereα is a positive number.
3.2.3. Step 1b: equation of motion with Verlet algorithm
Independently to the computation of the wave equation (33) and (34), we can solve the part of the system corresponding to the equation of motion. Find Uk ∈ R2∀k = 1, 2, ..., NGsuch as:
(ρ − ρ0) Sk dUk
dt · Vk− Fk· Vk = 0 ∀ Vk∈ R
2 (36)
We use the Molecular Dynamics algorithm [5, 6] to compute forces, integrate the equation of mo-tion for each grain and govern interacmo-tions between rigid grains. We choose to solve the equamo-tion of motion in three parts using the Verlet algorithm [30, 31].
1. First, position of the grain Xn+12
k is computed: Xn+ 1 2 k = X n k+ Δt U n k+ Δt 2 A n k (37)
2. Secondly, all forces F depending on the positions are computed:
An+12 k = F Xn+ 1 2 k (ρ − ρ0) Sk (38) 3. Finally, the velocity is deduced from the acceleration values:
Un+12 k = U n k+ Δt 2 An+12 k + A n k (39)
3.2.4. Step 2: enforcement of constraints with conjugate gradient algorithm
In the second time step of Marchuk’s operator splitting scheme, the jump condition of the acoustic wave equation is applied on the boundary ∂G and the rigid body motion constraint is applied inside the grain G. This leads to find un+1
h ∈ P 2 1h, U n+1 k ∈ R 2, λn+1 ∂Gh ∈ P1h andλ n+1 Gh ∈ P1h 11
such as: ⎧⎪⎪ ⎪⎪⎪⎪⎪ ⎪⎪⎪⎪⎪ ⎪⎪⎪⎪⎪ ⎪⎪⎪⎪⎪ ⎨ ⎪⎪⎪⎪⎪ ⎪⎪⎪⎪⎪ ⎪⎪⎪⎪⎪ ⎪⎪⎪⎪⎪ ⎪⎪⎩ Ωρ0 unh+1− un+ 1 2 h Δt · vhdx+ k ⎡ ⎢⎢⎢⎢⎢ ⎢⎢⎣(ρ − ρ0) Sk Unk+1− Un+ 1 2 k Δt · Vk ⎤ ⎥⎥⎥⎥⎥ ⎥⎥⎦ + k ∂Gk λn+1 ∂Gh(vh· nk− Vk· nk) d + G λn+1 Gh (∇ · vh) dx= 0 NGtimes ∂Gk unh+1· nk− Unk+1· nk μ∂Ghd = 0 G ∇ · un+1 h μGhdx= 0 (40a) (40b) (40c) ∀ vh∈ P21h, Vk ∈ R2,μ∂Gh ∈ P1h andμGh ∈ P1h.
The discretized system (40) can also be represented as matrices: ⎛ ⎜⎜⎜⎜⎜ ⎜⎜⎜⎜⎜ ⎜⎜⎜⎝ A1 0 CT1 CT3 0 A2 CT2 0 C1 C2 0 0 C3 0 0 0 ⎞ ⎟⎟⎟⎟⎟ ⎟⎟⎟⎟⎟ ⎟⎟⎟⎠ ⎛ ⎜⎜⎜⎜⎜ ⎜⎜⎜⎜⎜ ⎜⎜⎜⎝ un+1 Un+1 λ∂G λG ⎞ ⎟⎟⎟⎟⎟ ⎟⎟⎟⎟⎟ ⎟⎟⎟⎠ = ⎛ ⎜⎜⎜⎜⎜ ⎜⎜⎜⎜⎜ ⎜⎜⎜⎜⎝ un+12 Un+1 2 0 0 ⎞ ⎟⎟⎟⎟⎟ ⎟⎟⎟⎟⎟ ⎟⎟⎟⎟⎠ (41)
To simplify the notation, we introduce block matrices A, C; and vectors b, d, andλ:
A= A1 0 0 A2 ! C= C1 C2 C3 0 ! b= u n+1 Un+1 ! d= u n+12 Un+1 2 ! λ = λ∂G λG ! (42) and we rewrite system (41) as:
A CT C 0 ! b λ ! = d0 ! (43) Such system can be solved using Uzawa’s method for the conjugate gradient algorithm [32]. We initialize the algorithm with an initial approximationλ0and we construct an approximation of the
solution b for i= 0 until convergence by successively solving:
A bi+1 = d − CTλi (44)
and computing:
λi+1= λi+ β C bi+1 (45)
withβ a scalar.
The optimal rate of convergence of the algorithm is given by [33]:
β = 2
μ1+ μ2
(46) withμ1 andμ2the minimum and maximum eigenvalues of the Schur complement of sytem (43)
3.2.5. Velocity integration
Independently of Marchuk’s algorithm and after solving the system (40b), we integrate the final velocity Un+1
k of the grains to get their next position. We simply integrate the velocity for the whole time step to retrieve updated position of grains:
Xnk+1= Xnk+Δt
2
Unk+1+ Unk (47) Then the meshTGh is updated with the new positions X
n+1
k without modifying the regular mesh TΩh (Fig. 2).
4. Numerical experiments
In this part, we apply the method developed in sections 2 and 3 on three examples. On one hand, one movable grain is subject to an incident plane wave, and numerical results on dynamics and the acoustics are compared to analytic solutions. On the other hand two movable grains fixed by springs but without any incident plane wave illustrates energy transfers between fluid and grains. Finally we test a more realistic configuration of a random granular suspension with hundreds of grains to illustrate the possibilities of the model.
4.1. Movable grain in sound
To study the error compared to an analytic solution, we send a plane wave on a movable grain (Fig. 3). We first define a harmonic plane wave coming from the top (withθi= −π/2 the incident angle) and defined as:
pi(r, θ, t) = A ei(k r cos(θ+θi)− ω t) (48) with r the distance from the center of the grain and θ the observation angle. The pulsation ω and the wavenumber k are related to the frequency by f = 2ωπ = k c0
2π. The diameter of the grain
is d = 2 R = 1 mm. At the beginning of the experiment, the center of the grain is located at coordinates (5; 4.5) mm. The size of the domain Ω is 10 × 10 mm and the size of PML layers on the top and bottom is 5 hΩ. The source line is located at ysrc = 8.3 mm and there are PML layers
of thickness 0.4 mm on the top and on the bottom of the domain.
The wave velocity in the fluid medium is c0 = 1,500 m/s. The density of the fluid medium is
ρ0 = 1,000 kg/m3and the density of the material of grains is: ρ = 2,500 kg/m3. The frequency
of the incident wave is f = 1.5 MHz which corresponds to a of λ/d = 1. The amplitude of the incident wave is A= 1.5 Pa in terms of pressure or 1 μm/s in terms of velocity.
We discretize the domain Ω with N2 = 120 × 120 nodes and so h
Ω = 1/12 mm and the
maximum mesh size of the fictitious domains satisfy hG ≈ 1.2 √
2 hΩ. We set parameter α of Eq. (35) related to the CFL condition of Euler semi-implicit scheme for the time discretization to α = 0.65/√2. The duration of the simulation is tf = H/c0with H the height of the domainΩ.
0 2 4 6 8 10 X [ m m ] 0 2 4 6 8 10 Y [ mm ] fluid = 1,000 kg/m³ c = 1,500 m/s00 grain = 2,500 kg/m³ R = 0.5 mm source ysrc PML PML
Figure 3: Scheme of the experiment with 1 grain
4.1.1. Pressure and velocity fields
The reference analytic solution for the pressure field scattered by a movable rigid disc can be written in cylindrical coordinates as [34]:
ps(r, θ) =
∞
n=0
BnH(1)n (k r) cos (n (θ − θi)) ∀r ≥ R, 0 ≤ θ ≤ 2 π (49) with n an integer number and H(1)n is the Hankel function of the first kind. The incident pressure field can also be rewritten as a sum of Bessel functions of the first kind Jn:
pi(r, θ) = A ⎡ ⎢⎢⎢⎢⎢ ⎣J0(k r)+ 2 ∞ n=1 inJn(k r) cos (n (θ − θi)) ⎤ ⎥⎥⎥⎥⎥ ⎦ ∀r ≥ R, 0 ≤ θ ≤ 2 π (50)
Bnis a constant determined using Eqs. (49) and (50), and the boundary conditions applied on the grains defined in Eq. (3):
Bn= ⎧⎪⎪ ⎪⎪⎪⎪⎪ ⎪⎪⎪⎪⎪ ⎪⎪⎨ ⎪⎪⎪⎪⎪ ⎪⎪⎪⎪⎪ ⎪⎪⎪⎪⎩ −A J 0(k R) H(1)0 (k R) when n= 0 −2 A i k R J 1(k R)−ρ 0 ρ J1(k R) k R H(1)1 (k R)−ρ0 ρ H (1) 1 (k R) when n= 1 −2 A in J n(k R) H(1)n (k R) when n≥ 2 (51)
Then the total pressure field is the sum of the incident field piand the scattered field pr:
(a) Numerical solution (b) Analytic solution
Figure 4: Numerical solution of the total pressure field computed using the fictitious domain method (4a) versus the analytic solution (4b) for hΩ = 1/12 mm.
Fig. 4a shows the total pressure field computed with the fictitious domain method and Fig. 4b shows the analytic solution computed using Eqs. (49) and (50). When comparing figures 4a and 4b, we observe that the numerical solution presents good agreements with the analytic solution.
Our model also provides the acoustic velocity field. The analytic solution of the total velocity field can be deduced from the solution of the pressure field Eq. (52):
u(r, θ) = −i
ω c0
∇ p(r, θ) (53)
Figure 5 represents numerical and analytic solutions of the total velocity field. We can see that the two velocity fields are quite similar. The difference at the top of the domain is a numerical artifact due to the source line that generates a plane wave in two opposite directions.
We define the relative error eΦfor any fieldΦ during a period T as:
eΦ = sup tref≤t≤tref+T ⎛ ⎜⎜⎜⎜⎜ ⎝ Φnum Φ− Φan L2 an L2 ⎞ ⎟⎟⎟⎟⎟ ⎠ (54)
with tref = H/c0the time for the wave to propagate through the entire medium. The relative error
eΦis computed inside a square of size [1.5; 8.5] × [1; 8] mm to avoid PML and the effect of lateral periodic boundary conditions. Relative errors computed for fields p, ux and uy are reported in Table 1. The growth of the error on the acoustic fields is not optimal, revealing the weakness of Lagrangian finite elements when used in mixed methods [28]. The agreement of the numerical solution with the analytic solution (Figs. 5 and 4) shows nevertheless that the fictitious domain method converges with the element P1-P0on the contrary of the Raviart-Thomas element used in
[18].
(a) Numerical solution (b) Analytic solution
Figure 5: Numerical solution of the norm of the total velocity field computed using the fictitious domain method (5a) versus the analytic solution (5b) for hΩ = 1/12 mm.
Table 1: Relative errors computed for pressure and velocity fields for different discretizations
N2 hΩ ep eux euy 1202 1/12 mm 1.44 × 10−1 2.38 × 10−1 1.47 × 10−1 2402 1/24 mm 9.19 × 10−2 1.28 × 10−1 9.01 × 10−2
0 1 2 3 4 5 t[μs] −150 −100 −50 0 50 100 Uy [nm/s] N2=1202 N2=2402 N2=4802 (a) y-component 0 1 2 3 4 5 t[μs] −2.0 −1.5 −1.0 −0.5 0.0 0.5 1.0 1.5 2.0 2.5 Ux [nm/s] N2=1202 N2=2402 N2=4802 (b) x-component
Figure 6: Velocity of the grain function of the time computed using the fictitious domain method for several discretizations.
4.1.2. Velocity of the grain
The main feature of the model is to take into account the dynamic of the grain submitted to an incident wave. We plot on Figure 6 the velocity of the grain as a function of the time for three discretizations at t = 2.2 μs. We observe on Fig. 6a that grains start to oscillate at the same frequency as the incident wave, when the wavefront reaches its location.
The x-component of the velocity of the grain can also be assimilated to an absolute error because it should be Ux = 0 according to the analytic solution. So we note on Fig. 6b that the decrease of maximal error is better for the velocity of the grain than for the norm of the velocity field. Indeed, the scale of the grain is larger than the one of the finite elements and the grain is therefore less sensitive to small perturbations of the pressure field.
4.2. Two movable grains fixed by springs
We perform a second experiment to show that the model can describe acoustic emissions di-rectly coming from one grain subject to an external force Fk. Most numerical methods in acoustics can not work without an incident wave. In this experiment, we link each grain with its own spring of stiffness kG = 100 GN/m (to keep approximately the same frequency as in the previous example
f = √(k/m)/(2 π)), so that the external force Fkfrom Eq. (2) p. 3 becomes:
Fk= −kG
Yk− Ykeq
y (55)
with y the unitary vector of the y-axis and Ykeq the equilibrium position of the spring of the grain
k. The bottom grain has an initial displacement from equilibriumΔY0(t= 0) = +1 pm and the top
grain is at its equilibrium position (see scheme Fig. 7a). Both are put in the middle of a domain of size 15× 17 mm and at a distance of 1 mm of each other. We set PML on each side of the box
0 2 4 6 8 10 12 14 X [mm] 0 2 4 6 8 10 12 14 16 Y [mm] Y0 Grain k = 0 Grain k = 1
(a) Scheme of the experiment (b) Pressure field at t= 4.3 μs
Figure 7: Numerical experiment of two movable grains fixed by springs. Fig. 7a shows a simplified representation of the experiment and Fig. 7b the pressure field at the final time t= 4.3 μs.
but we stop the simulation before the wavefront reaches the layer. Figure 7b shows the numerical pressure field for a discretization step hΩ = 1/12 mm at the final time t = 4.3 μs.
We examine now to the energy inside the system which is composed, according to system (27), on one hand of acoustic energy inΩ :
Ea= Ω 1 2ρ0c20 p2dx+ Ω 1 2ρ0u 2dx (56)
and on the other hand of the mechanical energy of each grain k:
Emk = 1 2kG Yk− Ykeq 2 +1 2(ρ − ρ0) SkU 2 k (57)
Energies of the wave and the two grains are plotted in Figure 8. First we observe that the total energy in the system is well conserved considering the first order numerical scheme used for the time discretization. We observe that at t = 0, all the energy is in the grain 0 (in blue Fig. 8) in the form of mechanical potential energy due to the initial stretching of the spring. Then the grain 0 starts to oscillate and generates an acoustic field so its mechanical energy is partially converted to acoustic energy (in red Fig. 8). At t = 1 μs, the wavefront of the wave coming from the grain 0 reaches the other grain 1 and a part of the acoustic energy is converted to mechanical energy of grain 1 (in green Fig. 8). Then because the grain 1 is starting to oscillate, it is also radiating and gives back some mechanical energy in the form of acoustic energy.
As a comparison, we do the same experiment replacing water by air. Results reported in table 2 confirms that a numerical model able to link the wave equation and the equation of motion is essential to take in account transfers of energy through both water and grains. On the contrary, it is insignificant in the air.
0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0 t [μs] 0 10 20 30 40 50 60 E [fJ] Wave Grain k =0 Grain k =1
Figure 8: Evolution of the distribution of energy in the system containing two glass grains fixed by springs and surrounded by water
Table 2: Comparison of average energy distribution in water and in the air during the last period of oscillation water air c0 1,500 m/s 340 m/s ρ0 1,000 kg/m3 1.2 g/m3 Ea 95.9 % 0.6 % Em0 1.7 % 99.4 % Em1 2.4 % ≈ 0 % 19
0 5 10 15 20 x [mm] 0 10 20 30 40 y[mm] −2.4 −1.8 −1.2 −0.6 0.0 0.6 1.2 1.8 2.4 [Pa]
(a) Pressure field
−1 0 1 uy[μm/s] 0 5 10 15 20 x [mm] 0 10 20 30 40 y[mm] 0.00 0.25 0.50 0.75 1.00 1.25 1.50 1.75 2.00 [μm/s]
(b) Velocity field (uyand u )
0 5 10 15 20 X [mm] 0 10 20 30 40 Y [mm] U=0.30μm/s 0 80 160 240 320 400 480 560 |A| [mm/s^2] (c) Dynamics
Figure 9: Propagation of a pulse through a granular suspension at t = 20 μs. Fig. (a) shows the acoustic pressure field, Fig. (b) the mean of the y-component and the norm of the velocity field and Fig. (c) positions, velocities (represented by red arrows) and acceleration (in grayscale) of each grain.
4.3. Acoustic pulse propagation through a granular suspension
The last experiment involves a random suspension of 400 movable grains subject to a unique pulse. We show that the model is able to work with many grains, contrary to the one developed in [18]. The source is on a horizontal line located at ysrc= 4.4 cm and the pulse is a the 4th derivative
of a Gaussian with the central frequency fc = 150 kHz. We choose a pulse to follow easily the propagation of the wave through the medium. We consider a case where the characteristic wavelengthλ/d = 10 because displacement of grains is maximum at λ/d 1. The domain Ω is a rectangle 2 cm width and 4.8 cm height discretized by Nx× Ny= 108 × 260 and so hΩ = 5/27 mm. The boundary conditions are the same as in section 4.1: PML on the top and on the bottom; and periodic boundary conditions on the left side and right sides.
Fig. 9 show snapshots of the pressure field (a), the velocity field (b) and positions, velocities and acceleration of each grain (c). The thin figure between the velocity field snapshot and the pressure field snapshot shows the mean value of the y-component of the velocity field and helps the reader to follow the propagation of the pulse. The variations of pressure (Fig. 9a) around the grains determine the value of the hydrodynamic force acting on them. Since the hydrodynamic force is the unique force, its value is proportional to the acceleration A (in gray scale Fig. 9c).
Moreover, by observing red arrows in Fig. 9c, we can see that the wave sets in motion grains under its acoustic field. The result of the ratio between the amplitude of the velocity of one grain and the amplitude of the velocity field is provided by the following 2D analytic solution for low wavelength (inspired from the 3D version in [35]) :
U u =
2ρ0
ρ + ρ0 ≈ 0.57
Comparison of Figs. 9b and 9c shows that numerical results are similar to the one predicted by the analytic solution.
Then, we can see Fig. 10 that the pulse propagates through the granular suspension from the top to the bottom. The mean value of the amplitude the velocity field Fig. 10b shows that the amplitude of the wave is decreasing as the energy of the wave is transmitted to the grains Fig. 10c. We also observe on Fig. 10a several other reflections after the wave pass. Thus, we can use solution (58) only at the beginning of the propagation when grains only subject to the incident wave not yet reflected waves from their neighbors.
5. Conclusions and discussion
Our results show that this fictitious domain method can couple the acoustic wave equation and the equation of motion for rigid grains. We show this using three numerical experiments. First, both acoustic fields and dynamics of grains computed using this model satisfy the analytical solution of a plane wave diffracted by a movable grain. In the second case, energy is transmitted between two grains and the fluid without loss of total energy. The last experiment shows that this method can also be expanded to hundreds of grains and used in the framework of acoustic experiments on granular suspensions.
The strength of this model is the ability of each grain to generate its own acoustic field when subject to an external force or due to an incident wave. As far as we know, there are no other acoustic model capable of coupling dynamics of rigid grains and waves. The rigidity of grains im-plies an infinite velocity inside the grains but still a finite velocity through the contacts network of a granular packing. In our knowledge, our model is the only one able to record acoustic emissions related to any forces acting on the grains of a granular medium.
Nevertheless interesting comparative studies could be done by observing changes in the re-flected wave signal while changing parameters related to contact forces or materials of the grains. In the particular framework of compact granular media, interaction forces are defined according to linear or elastic models to simulate repulsion and friction between grains. So waves generated by grains are then directly related to the parameters of these models. Therefore we plan to use our model to study acoustic waves emitted by a granular medium during a gravitational destabilization.
References
[1] Y. Le Gonidec, D. Gibert, Multiscale analysis of waves reflected by granular media: Acoustic experi-ments on glass beads and effective medium theories, J. Geophys. Res. - Sol. Ea. 112 (B5) (2007) B05103. doi:10.1029/2006JB004518.
[2] V. Y. Zaitesev, P. Richard, R. Delannay, V. Tournat, V. E. Gusev, Pre-avalanche structural rearrangements in the bulk of granular medium: Experimental evidence, EPL 83 (6) (2008) 64003. doi:10.1209/0295-5075/83/64003. [3] J.-L. Thirot, Y. Le Gonidec, B. Kergosien, Acoustic emissions in granular structures under gravitational destabi-lization, in: B. B. J. Linde, J. Paczkowski, N. Ponikwicki (Eds.), Acoustics of ordered and disordered granular strctures, Vol. 1433 of AIP Conference Proceedings, International Congress on Ultrasonics 2011, AIP, Gda´nsk, Poland, 2012, pp. 143–146. doi:10.1063/1.3703157.
[4] V. Tournat, V. Y. Zaitsev, V. E. Gusev, V. Nazarov, P. B´equin, B. Castagn`ede, Probing weak forces in granular media through nonlinear dynamic dilatancy: clapping contacts and polarization anisotropy, Phys. Rev. Lett. 92 (2004) 085502. doi:10.1103/PhysRevLett.92.085502.
0 5 10 15 20 x [mm] 0 10 20 30 40 y[mm] −2.4 −1.8 −1.2 −0.6 0.0 0.6 1.2 1.8 2.4 [Pa] −1 0 1 uy[μm/s] 0 5 10 15 20 x [mm] 0 10 20 30 40 y[mm] 0.00 0.25 0.50 0.75 1.00 1.25 1.50 1.75 2.00 [μm/s] 0 5 10 15 20 X [mm] 0 10 20 30 40 Y [mm] U=0.36μm/s 0 80 160 240 320 400 480 560 640 720 |A| [mm/s^2] 0 5 10 15 20 x [mm] 0 10 20 30 40 y[mm] −2.4 −1.8 −1.2 −0.6 0.0 0.6 1.2 1.8 2.4 [Pa] −1 0 1 uy[μm/s] 0 5 10 15 20 x [mm] 0 10 20 30 40 y[mm] 0.00 0.25 0.50 0.75 1.00 1.25 1.50 1.75 2.00 [μm/s] 0 5 10 15 20 X [mm] 0 10 20 30 40 Y [mm] U=0.31μm/s 0 100 200 300 400 500 600 700 800 |A| [mm/s^2] 0 5 10 15 20 x [mm] 0 10 20 30 40 y[mm] −3.2 −2.4 −1.6 −0.8 0.0 0.8 1.6 2.4 3.2 [Pa]
(a) Pressure field
−1 0 1 uy[μm/s] 0 5 10 15 20 x [mm] 0 10 20 30 40 y[mm] 0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4 1.6 1.8 [μm/s]
(b) Norm of the velocity field
0 5 10 15 20 X [mm] 0 10 20 30 40 Y [mm] U=0.41μm/s 80 160 240 320 400 480 560 640 |A| [mm/s^2] (c) Dynamics
[5] P. A. Cundall, O. D. L. Strack, A discrete numerical model for granular assemblies, G´eotechnique 29 (1) (1979) 47–65. doi:10.1680/geot.1979.29.1.47.
[6] S. McNamara, M´ethode Dynamique Mol´eculaire, in: F. Radja¨ı, F. Dubois (Eds.), Mod´elisation num´erique discr`ete de mat´eriaux granulaires, Trait´e MIM - M´ecanique et ing´enierie des mat´eriaux, Herm`es-Lavoisier, 2010, Ch. 1, pp. 25–48.
[7] R. M. Alford, K. R. Kelly, D. M. Boore, Accuracy of finite-difference modeling of the acoustic wave equation, Geophysics 39 (6) (1974) 834–842.
[8] D. Kosloff, D. Kessler, Seismic numerical modeling, in: Y. Desaubies, A. Tarantola, J. Zinn-Justin (Eds.), Oceanographic and Geophysical Tomography, Houches Summer Session, Elsevier Science, Les Houches, France, 1990, pp. 249–312.
[9] O. C. Zienkiewicz, The finite element method in engineering science, McGraw-Hill, 1971.
[10] I. Babuˇska, The finite element method with Lagrange multipliers, Numer. Math. 20 (3) (1973) 179–192. doi:10.1007/BF01436561.
[11] R. Glowinski, T. Pan, J. Periaux, A Lagrange multiplier/ fictitious domain method for the Dirichlet problem -generalization to some flow problems, Jpn J. Ind. Appl. Math 12 (1) (1995) 87–108. doi:10.1007/BF03167383. [12] E. Heikkola, Y. A. Kuznetsov, P. Neittaanm¨aki, J. Toivanen, Fictitious domain methods for the nu-merical solution of two-dimensional scattering problems, J. Comput. Phys. 145 (1) (1998) 89–109. doi:10.1006/jcph.1998.6014.
[13] R. Glowinski, T. Pan, J. Periaux, Fictitious domain methods for incompressible viscous flow around moving rigid bodies, in: J. R. Whiteman (Ed.), The Mathematics of Finite Elements and Applications, 1996 MAFELAP Conference, Whiley, Brunel University, 1997.
[14] L. Rhaouti, A. Chaigne, P. Joly, Time-domain modeling and numerical simulation of a kettledrum, J. Acoust. Soc. Am. 105 (6) (1999) 3545–3562. doi:10.1121/1.424679.
[15] R. Glowinski, Y. Kuznetsov, On the solution of the Dirichlet problem for linear elliptic operators by a dis-tributed Lagrande multiplier method, CR Acad. Sci. I-Math. 327 (7) (1998) 693–698. doi:10.1016/S0764-4442(99)80103-7.
[16] R. Glowinski, T. Pan, T. Hesla, D. Joseph, A distributed Lagrange multiplier/ fictitious domain method for particulate flows, Int. J. Multiphase Flow 25 (5) (1999) 755–794. doi:10.1016/S0301-9322(98)00048-2. [17] V. Bokil, R. Glowinksi, An operator splitting scheme with a distributed Lagrange multiplier based
fic-titious domain method for wave propagation problems, J. Comput. Phys. 205 (1) (2005) 242–268. doi:10.1016/j.jcp.2004.10.040.
[18] D. Imbert, S. McNamara, Fictitious domain method to model a movable rigid body in a sound wave, J. Numer. Math. 20 (3–4) (2012) 267–285. doi:10.1515/jnum-2012-0014.
[19] J.-P. Berenger, A perfectly matched layer for the absorption of electromagnetic waves, J. Comput. Phys. 114 (2) (1994) 185–200. doi:10.1006/jcph.1994.1159.
[20] W. C. Chew, W. H. Weedon, A 3D perfectly matched medium from modified Maxwell’s equations with stretched coordinates, Microw. Opt. Techn. Let. 7 (13) (1994) 599–604. doi:10.1002/mop.4650071304.
[21] H. H. Hu, Direct simulation of flows of solid-liquid mixtures, Int. J. Multiphase Flow 22 (2) (1996) 335–352. doi:10.1016/0301-9322(95)00068-2.
[22] B. Lombard, Mod´elisation num´erique de la propagation des ondes acoustiques et ´elastiques en pr´esence d’interfaces, Phd thesis, Universit´e de la M´editerran´ee, 2002.
[23] N. A. Patankar, P. Singh, D. D. Joseph, R. Glowinski, T.-W. Pan, A new formulation of the distributed Lagrange multiplier/ fictitious domain method for particulate flow, Int. J. Multiphase Flow 26 (9) (2000) 1509–1524. doi:10.1016/S0301-9322(99)00100-7
[24] R. Glowinski, T. Pan, J. P´eriaux, Distributed Lagrange multiplier methods for incompressible viscous flow around moving rigid bodies, Comput. Methods Appl. Mech. Engrg. 151 (1–2) (1998) 181–194. doi:10.1016/S0045-7825(97)00116-3.
[25] F. Hecht, FreeFem++, Laboratoire Jacques-Louis Lions, Universit´e Pierre et Marie Curie, Paris, 3rd Edition (2012).
URL http://www.freefem.org/ff++/ftp/freefem++doc.pdf
[26] P. A. Raviart, J. M. Thomas, A mixed finite element method for 2-nd order elliptic problems, in: I. Galligani,
E. Magenes (Eds.), Mathematical Aspects of Finite Element Methods, Vol. 606 of Lecture Notes in Mathematics, Springer, Rome, Italy, 1977, pp. 292–315. doi:10.1007/BFb0064470.
[27] E. B´ecache, J. Rodr´ıguez, C. Tsogka, Convergence results of the fictitious domain method for a mixed formu-lation of the wave equation with a Neumann boundary condition, ESAIM Math. Model. Num. 43 (2) (2009) 377–398. doi:10.1051/m2an:2008047.
[28] J. Li, T. Arbogast, Y. Huang, Mixed methods using standard conforming finite elements, Comput. Methods Appl. Mech. Engrg. 198 (5–8) (2009) 680–692. doi:10.1016/j.cma.2008.10.002.
[29] G. I. Marchuk, Splitting and alternating direction methods, Vol. 1 of Handbook of Numerical Analysis, Elsevier, 1990. doi:10.1016/S1570-8659(05)80035-3.
[30] L. Verlet, Computer ”experiments” on classical fluids. I. Thermodynamical properties of Lennard-Jones molecules, Phys. Rev. 159 (1) (1967) 98–103. doi:10.1103/PhysRev.159.98.
[31] W. C. Swope, H. C. Andersen, H. Berens, K. R. Wilson, A computer simulation method for the calculation of equilibrium constants for the formation of physical clusters of molecules: Application to small water clusters, J. Chem. Phys. 76 (1) (1982) 637–649. doi:10.1063/1.442716.
[32] K. J. Arrow, L. Hurwicz, H. Uzawa (Eds.), Studies in linear and non-linear programming, Vol. 2 of Stanford Mathematical Studies in the Social Sciences, Stanford University Press, 1958.
[33] H. C. Elman, G. H. Golub, Inexact and preconditioned Uzawa algorithms for saddle point problems, SIAM J. Numer. Anal. 31 (6) (1994) 1645–1661. doi:10.1137/0731085.
[34] C. Potel, M. Bruneau, Acoustique G´en´erale - ´Equations diff´erentielles et int´egrales, solutions en milieux fluide et solide, applications, Technosup, Ellipse, 2006.
